The Terrors, who have been relegated from the Premiership, were punished by the governing body after fielding ineligible players Jamie Robson and Alistair Coote in last Friday's match against Inverness Caledonian Thistle.
Robson was an unused substitute, while Coote came off the bench in the 76th minute of the game, which ended in a 3-2 victory for United.
As a result of breaching SPFL rules, the club were docked three points of their overall total this season and hit with a £30,000 fine, which has been partially suspended.
In response, the club confirmed that Thompson has stepped down from his position on the SPFL board. A statement via the club's official website read: "[Thompson] has expressed his shock and anger at the severity of the sanction imposed by the SPFL in relation to a rule breach in our recent Ladbrokes Premiership fixture versus Inverness Caley Thistle.
"Whilst acknowledging the club's error and accepting a breach of rules occurred, the chairman feels the gravity of the punishment is wholly disproportionate for what was a simple honest, human error. We await the full written judgment after which we will consider our next action.
"Our travelling fans are the ones who have been punished the most here and the club will examine ways to make this up to them."
United won just seven of their 37 games this season.